Given the simplest system containing one bosonic and one fermionic degree of freedom with the Hilbert space spanned by
\begin{align}
|n,m\rangle\quad\text{with}\quad n\in\mathbb{N}\quad\text{and}\quad m\in\{0,1\}\,,
\end{align}
such that $\hat{a}$ and $\hat{a}^\dagger$ are bosonic creation and annihilation operators for the occupation number $n$ and $\hat{b}$ and $\hat{b}^\dagger$ are fermionic creation and annihilation operators for the occupation number $m$. This means we have
\begin{align}
[\hat{a},\hat{a}^\dagger]=1\quad\text{and}\quad[\hat{b},\hat{b}^\dagger]_+=1\,.
\end{align}
I was wondering if there is a Bogoliubov-like transformation that mixes fermionic and bosonic operators to find a new operators
\begin{align}
\tilde{a}&=A\,a+B\,a^\dagger+C\,b+D\,b^\dagger\,\\
\tilde{b}&=E\,a+F\,a^\dagger+G\,b+H\,b^\dagger\,\\
\end{align}
such that we can build a new Fock space on it with new vectors
\begin{align}
|\tilde{n},\tilde{m}\rangle\,.
\end{align}
Can you recommend a reference on this? In particular, I read a lot about ``complex structures'' that one can use to parametrize the different choices of raising & lowering operators and would be interested if one can also apply this framework in the supersymmetric case.

This post imported from StackExchange Physics at 2016-08-13 17:01 (UTC), posted by SE-user LFH